# How is there a change in magnetic flux from a moving conductor through a uniform magnetic field?

My textbook states that when a straight conductor with a velocity perpendicular to itself and the magnetic field (see image), an emf will be induced between the two ends of a conductor. This means that there is a change in magnetic flux, but how is that so? According to the definition of magnetic flux = B*A, there is no change in either the strength of a magnetic field or the area perpendicular as the conductor moves through the magnetic field.

Could someone please let me understand why this is incorrect.

straight conductor moves perpendicular to itself

I don't quite get what you mean by this. But I think I understood the scope of your question.

In an ideal conductor it has an "infinite" number of free charges that can move. When a charge moves in the presence of a magnetic field it experiences a force given by $$\vec{F_{mag}}^{\,}=q\times\vec{B} \, \,$$, so this causes the the free charges to move (and this example it happens that the orientations will have the free positive charges move upward to one end). When the free charges do move, you have a difference in potential at the ends of the rod which qualifies as EMF.

• Thanks for your response, I understand your reasoning here, which qualifies that an EMF is induced. However, as an EMF is induced, that means there is a change of magnetic flux. So how is there a change in magnetic flux if the area of the conductor and strength of the magnetic field are constant? Jun 16, 2019 at 0:28
• A moving charge generates a magnetic field, so when the free charges of the conductor move they generate a magnetic field, and since they are moving the magnetic field they generate moves with them, hence a change in magnetic flux. Jun 16, 2019 at 0:43
• I see that! Books gloss over those details. Jun 16, 2019 at 2:04
• This also begs the question then: if a conducting loop is considered instead of a straight conductor, will there be an induced emf/change in magnetic flux due to the magnetic force applied to the free charges within it. Jun 16, 2019 at 2:10

There is no change in magnetic flux, as you've said. What you're missing is, there is also no EMF induced in any loop. Try drawing a loop you think there is an EMF around. If a positive charge starts at the top of the conductor, goes to the bottom of the conductor, and then goes back to the top, it doesn't gain or lose kinetic energy. I think your confusion can be cleared up by remembering what the Biot-Savart law actually says: It says that changing magnetic flux through a loop results in an EMF around the loop. You've said there's no changing magnetic flux through any loop you can draw, and if you think about it there is also no EMF around any loop you can draw.

• this is conflicting with what the other answer says? Jun 16, 2019 at 12:34
• @Max604 I don't agree with the other answer. This problem just has nothing to do with magnetic flux because it has nothing to do with emf around a loop. Jun 16, 2019 at 15:26